\newproblem{lay:5_5_25}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.5.25}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $A$ be a real $n\times n$ matrix, and let $\mathbf{x}\in\mathbb{C}^n$. Show that $\mathrm{Real}\{A\mathbf{x}\}=A\mathrm{Real}\{\mathbf{x}\}$ and 
	$\mathrm{Imag}\{A\mathbf{x}\}=A\mathrm{Imag}\{\mathbf{x}\}$.
}{
  % Solution
	Consider
	\begin{center}
		$\mathbf{x}=\mathrm{Real}\{\mathbf{x}\}+i\mathrm{Imag}\{\mathbf{x}\}$\\
	\end{center}
	Multiplying on both sides by $A$ on the left
	\begin{center}
		$\begin{array}{rcl}
		   A\mathbf{x}&=&A(\mathrm{Real}\{\mathbf{x}\}+i\mathrm{Imag}\{\mathbf{x}\})\\
			   &=&A\mathrm{Real}\{\mathbf{x}\}+iA\mathrm{Imag}\{\mathbf{x}\}
		\end{array}$
	\end{center}
	Now simply by taking the real and imaginary parts of $A\mathbf{x}$ and taking into account that $A$ is a real matrix, we get the properties proposed:
	\begin{center}
		$\begin{array}{rcl}
		   \mathrm{Real}\{A\mathbf{x}\}&=&A\mathrm{Real}\{\mathbf{x}\}\\
		   \mathrm{Imag}\{A\mathbf{x}\}&=&A\mathrm{Imag}\{\mathbf{x}\}\\
		\end{array}$
	\end{center}
}
\useproblem{lay:5_5_25}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
